Extended discussion

Spoilers ahead! You have been warned. I'll start by providing answers for
the clues, and then go on to reveal some other things that you might want to
figure out on your own. So scroll wisely.

The clues

First of all, if you couldn't make heads or tails of any of the clues, you
need to understand how cryptic clues
work. Read up on that and try again. Solving cryptic crosswords is a highly
rewarding pastime, so it will be worth the effort.

Made from wings I've lost in dreams.

Definition: Lost in dreams. Wordplay: Made from wings = pens, literally “I've”. Pensive.

(This clue used to read “Wings I've lost in dreams”, but I decided to change it to make it more fair. I didn't change the title of the cryptic.)

The German random number generator.

Looked up “copulative”.

Structure

Figure 2: The first 2500 steps of Rule 124 (only the leftmost 200 columns are shown).

Wings I've lost in dreams features an infinitely large grid that can be regarded as
an array of macro blocks, eight by six squares in size:

Figure 1: A macro
block

Each macro block has an output, indicated by a star in Figure 1 above. The
output is either I or O, and thus represents one bit of
information.

The seven-letter horizontal word along the bottom of the macro
block—we'll refer to it as the pivotal word—determines the
output based on its first, third and seventh letter. Of all the seven-letter
words given by the clues, there are exactly eight that can fit in this slot;
one for each possible combination of three binary inputs.

The inputs of a macro block correspond to the outputs of the three blocks
above it, i.e. the neighbours to the north-west, north and north-east. This
information is routed by means of intermediary word slots, each capable of
holding one of a pair of words. In this way, information flows downwards
through the grid. The I or O at an output selects one of two
possible five-letter words going down. Each of these words fully determines the
two horizontal seven-letter words that intersect it, and these carry the
information further towards the pivotal words of the appropriate neighbouring
macro blocks. In one place, marked with a dot in Figure 1, two words cross
without influencing each other. This was achieved by ensuring that all words
that may appear in these slots have the same letter (an O) at the point
of intersection.

The following table summarises the relation that is enforced between the
inputs and outputs of each macro block. The order of the columns in the table
corresponds to the order of the letters within the pivotal words, i.e. the output
is given in the third column from the left.

NW

N

Out

NE

0

0

0

0

0

0

0

1

0

1

1

0

0

1

1

1

1

0

1

0

1

0

1

1

1

1

1

0

1

1

0

1

Table 1: The update rule

Cellular automaton interpretation

This array of macro blocks behaves like a cellular automaton.
Each row of cells corresponds to the state of the
automaton at a given moment, and is comprised of an infinite number of binary
digits. Each row of macro blocks fully determines what words can go into the
row below it, just like the state of a cellular automaton fully determines the
next state. The update rule for the automaton is determined by the set
of pivotal words provided by the clues. The clues of this cryptic have been
chosen in such a way that the state of the cellular automaton is updated
according to Rule 124 (which is the number obtained, in binary, from the
output column of Table 1 in bottom-up order). This is a mirror version of
the more well-known Rule 110.

The initial state (or configuration) of the automaton is determined
by the Os at the top of the grid. The left edge of the grid has been
crafted to make the north-west neighbour of every leftmost cell be an
I-block. As it happens, these conditions trigger a long and
beautifully complex sequence of state interactions in the Rule 124
automaton, shown in Figure 2, in which the pattern comprised of the
provided ANSWER letters appears exactly once along the left edge of
the grid (indicated by a small arrow).

Obtaining the final answer

It should be mentioned at this point that the ANSWER condition
appears after 2149 state transitions. In order to obtain the final answer
in the traditional way, one would have to pencil in over fifteen million
words. At one word per second, working eight-hour days, that'd still take
roughly a year and a half. Please be advised that you can solve the problem
considerably faster by implementing a computer program that simulates the
cellular automaton—even if that involves learning how to program a
computer in the first place. Or, if you still trust me, you could just zoom in
on Figure 2.

You may have tried to work backwards from the ANSWER letters in
order to figure out what letters can possibly appear in the grey squares. Due
to the complex behaviour of Rule 124, it is generally not possible to go
back in time and determine an earlier state vector; one has to run the
automaton forwards.

Based on our knowledge of how the macro blocks work, we can nevertheless
reduce the number of possible combinations of letters, but this will not be
enough. The grey squares can in fact accomodate two different six-letter words,
each of which is the name of a pioneer in Computer Science. A famous thesis
relavant to the present discussion has been named after these two pioneers. But
in order to determine which of the two names is the final answer, you're going
to have to run the automaton.

Universality

It has
been proved that Rule 110 (and thus Rule 124 by symmetry) is Turing complete.
This means that the cellular automaton is capable of carrying out
computations—in fact, that it can evaluate any computable
function.

All you need to do is encode the desired computation and its input in the
initial state vector, which can be an arbitrarily large binary number padded
with an infinite number of zeros to the right. Then you run the automaton forwards
until the ANSWER condition (or some similar condition of your choice)
is true, at which point you read out the solution. The details of how you would
represent the computation using a bit vector are provided in the aforementioned
proof.

With the given ANSWER condition, the function can only produce one
bit of output. However, it is always possible to represent an integer-valued
function F(x) as a boolean-valued function F'(x,
b) that computes F(x) and then extracts bit b of
the result.

Now, suppose you had been asked to solve this cryptic crossword without
being restricted to a particular initial state, i.e. without having any
O letters already filled-in. This version of the cryptic would
have many possible solutions, and would, by virtue of being structurally
equivalent to the Rule 124 cellular automaton, be a universal Turing
machine. So in principle, when you scratch out the Os, Wings
I've lost in dreams generalises into a fully functional computer, capable
of running any software by emulation. You could run a crossword solver on it,
for instance, or a program that simulates cellular automata.

None of this is practical. Even the most trivial of programs would require
initial state vectors spanning millions of bits. But it is possible. And
so, what you have seen turns out to be the world's first cryptic crossword that
is capable of solving itself.

Posted Wednesday 6-Jul-2016 23:53

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Fantastic. (I haven’t solved it, but did figure out all clues except one and understood the construction.)

I didn’t like PENS for WINGS.It’s not abundantly clear if LOOP or POOL is the answer (because “over” could modify both before and after).The Eve clue is way too hard and doesn’t even contain a straight definition.

—Thore

AnonymousThu 7-Jul-2016 13:18

Oh, and some solvers dislike removal of an anagram (as in ALE*). According to that school, subtracted letters must be present in the right order.

lftLinus ÅkessonThu 7-Jul-2016 22:37

Your first objection is a matter of a pinion (ba-dum tsch!), but I've updated the clue to make it more fair. And you're right about Eve, so I turned her into an ornament. The loop/pool ambiguity is duly noted, but the matter is resolved by the crossing words, so I'll leave that the way it is. Same with the regurgitated ale; I've learned something, but I don't think it's necessary to change the clue, especially if that's not a universal rule.